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[数学文化，客观派，讨论] 欧几里得对“素数有无穷多个”研究的有效性

已有 1225 次阅读 2024-11-17 22:51 |个人分类:基础数学-逻辑-物理|系统分类:科研笔记

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[数学文化，客观派，讨论] 欧几里得对“素数有无穷多个”研究的有效性

数学： mathematics

数学基础： foundations of mathematics

素数： prime number

素数的数量是无限的： there are an infinite number of primes

素数的集合是无限的： collection of primes is infinite

欧几里得： Euclid, Eukleides

要点：

讨论欧几里得对“素数有无穷多个”研究的有效性。

Euclid-woodcut-1584 britannica_小.jpg

欧几里得 Euclid, Eukleides, 活动于约公元前300年

https://cdn.britannica.com/46/8446-050-BC92B998/Euclid-woodcut-1584.jpg

https://www.britannica.com/biography/Euclid-Greek-mathematician

一、讨论：欧几里得对“素数有无穷多个”研究的有效性

1.1 欧几里得的证明

公元前 3世纪，欧几里得（Euclid, Eukleides）证明：

已知素数的连乘积加1，是“新素数”或“以新素数为因子”。

Books VII–IX contain elements of number theory, where number (arithmos) means positive integers greater than 1. Beginning with 22 new definitions—such as unity, even, odd, and prime—these books develop various properties of the positive integers. For instance, Book VII describes a method, antanaresis (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers; Book VIII examines numbers in continued proportions, now known as geometric sequences (such as ax, ax², ax³, ax⁴…); and Book IX proves that there are an infinite number of primes.

https://www.britannica.com/biography/Euclid-Greek-mathematician

https://www.britannica.com/science/number-theory/Euclid

Third, Euclid showed that no finite collection of primes contains them all. His argument, Proposition 20 of Book IX, remains one of the most elegant proofs in all of mathematics. Beginning with any finite collection of primes—say, a, b, c, …, n—Euclid considered the number formed by adding one to their product: N = (abc⋯n) + 1. He then examined the two alternatives:

(1) If N is prime, then it is a new prime not among a, b, c, …, n because it is larger than all of these. For example, if the original primes were 2, 3, and 7, then N = (2 × 3 × 7) + 1 = 43 is a larger prime. (2) Alternately, if N is composite, it must have a prime factor which, as Euclid demonstrated, cannot be one of the originals. To illustrate, begin with primes 2, 7, and 11, so that N = (2 × 7 × 11) + 1 = 155. This is composite, but its prime factors 5 and 31 do not appear among the originals. Either way, a finite set of primes can always be augmented. It follows, by this beautiful piece of logic, that the collection of primes is infinite.

https://www.britannica.com/science/number-theory/Euclid

1.3 讨论：欧几里得对“素数有无穷多个”研究的有效性

欢迎老师们发言！

二、相关的逻辑问题

2.1 演绎证明的结论，是前提（假设）蕴含的

一个现存的解释是，演绎推理的结论早已包含在前提之中了，实际上是已知的，推理过程只不过是把前提中隐含的信息明朗化，是对前提中已有内容的某种重复。因此，演绎推理推不出新知识。

吴炜，程本学，李珍编著. 自然辩证法概论[M]. 2019，第 122-123 页

通俗些：

采用演绎推理，您的“前提”假定了什么，经过“严格”推理（以及符号公式推导）之后，就会得到“您假定的结果”。

严格说，演绎证明差不多几乎就是“同义反复”。

2.2 完全归纳推理不属于归纳推理范围，属于演绎推理的范围

2022-01-20，完全归纳推理/complete inductive inference/张晓芒，中国大百科全书，第三版网络版[DB/OL]

https://www.zgbk.com/ecph/words?SiteID=1&ID=118651&Type=bkzyb&SubID=104156

归纳推理的极限形式。又称完全归纳法。

完全归纳推理由前提到结论之间是一种必然性推理，所以现代逻辑学认为完全归纳推理不属于归纳推理范围，属于演绎推理的范围，是演绎推理的逆向归纳法。

2.3 《Encyclopedia of Mathematics》的词条“Proof”

https://encyclopediaofmath.org/wiki/Proof

Proof

A reasoning conducted according to certain rules in order to demonstrate some proposition (statement, theorem); it is based on initial statements (axioms). In practice, however, it may also be based on previously demonstrated propositions. Any proof is relative, since it is based on certain unprovable assumptions. Rules of conducting a reasoning and methods of proof form a main topic in logic. See Proof theory.

【机器翻译】证明

根据某些规则进行的推理，以证明某个命题（陈述、定理）；它基于初始陈述（公理）。然而，在实践中，它也可能基于之前证明的命题。任何证明都是相对的，因为它是基于某些未经证明（unprovable）的假设。推理规则和证明方法是逻辑学的一个主要课题。参见证明论。

参考资料：

[1] 卢昌海. 素数有无穷多个之九类证明[J]. 数学文化，2018, 9(4): 73-84.

https://www.global-sci.org/intro/article_detail/mc/12858.html

https://www.global-sci.org/intro/articles_list/mc/1488.html

[2] Prime number. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Prime_number

The prime numbers play the role of "construction blocks" from which one can construct all other natural numbers. Already in the 3rd century B.C., Euclid proved that the set of prime numbers is infinite, and Eratosthenes found a way of sifting the prime numbers out of the sequence of natural numbers (cf. Eratosthenes, sieve of).

[3] Euclid, Greek mathematician, britannica

https://www.britannica.com/biography/Euclid-Greek-mathematician

Books VII–IX contain elements of number theory, where number (arithmos) means positive integers greater than 1. Beginning with 22 new definitions—such as unity, even, odd, and prime—these books develop various properties of the positive integers. For instance, Book VII describes a method, antanaresis (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers; Book VIII examines numbers in continued proportions, now known as geometric sequences (such as ax, ax2, ax3, ax4…); and Book IX proves that there are an infinite number of primes.

[4] Proof. A.S. Kuzichev (originator), Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Proof

[5] prime number theorem, mathematics, britannica

https://www.britannica.com/science/prime-number-theorem

[6] fundamental theorem of arithmetic, mathematics, britannica

https://www.britannica.com/science/fundamental-theorem-of-arithmetic

Also known as: theorem of prime factorization, unique factorization theorem

fundamental theorem of arithmetic, Fundamental principle of number theory proved by Carl Friedrich Gauss in 1801. It states that any integer greater than 1 can be expressed as the product of prime numbers in only one way.

[7] number theory, mathematics, britannica

https://www.britannica.com/science/number-theory/Euclid

[8] Riemann zeta function, mathematics, britannica

https://www.britannica.com/science/Riemann-zeta-function

[9] Riemann zeta function. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Zeta-function#Riemann.27s_zeta-function

[10] 科普中国，2021-12-31，-科学百科知识第7期丨什么是素数？数学家为什么对它们感兴趣？

https://www.kepuchina.cn/article/articleinfo?business_type=100&classify=0&ar_id=263252

[11] 科普中国，2021-12-31，素数定理

https://www.kepuchina.cn/article/articleinfo?business_type=100&classify=0&ar_id=289518

[12] 科普中国，2021-12-31，最大素数有用吗？安全上网就靠它

https://www.kepuchina.cn/article/articleinfo?business_type=100&classify=0&ar_id=177554